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Common data for Questions 50 and 51: Given $f(t)$ and $g(t)$ as shown below. The Laplace transform of $g(t)$ is

1. $\frac{1}{s}(e^{-3s}-e^{-5s})$
2. $\frac{1}{s}(e^{-s}-e^{-3s})$
3. $\frac{e^{-3s}}{s}(1-e^{-2s})$
4. $\frac{1}{s}(e^{-2s}-e^{-3s})$

Correct answer: $\frac{e^{-3s}}{s}(1-e^{-2s})$

Solution

The given signal is a rectangular pulse starting at $t=3$ and ending at $t=5$. Such a pulse can be written as $u(t-3)-u(t-5)$, whose Laplace transform is $\frac{e^{-3s}-e^{-5s}}{s} = \frac{e^{-3s}}{s}(1-e^{-2s})$.

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